Editing Holonomy (section)

===Holonomy of a connection in a vector bundle===
Let ''E'' be a rank-''k'' [[vector bundle]] over a [[smooth manifold]] ''M'', and let ∇ be a [[connection (vector bundle)|connection]] on ''E''. Given a [[piecewise]] smooth [[loop (topology)|loop]] ''γ'' : [0,1] → ''M'' based at ''x'' in ''M'', the connection defines a [[parallel transport]] map ''P''<sub>''γ''</sub> : ''E<sub>x</sub>'' → ''E<sub>x</sub>'' on the fiber of ''E'' at ''x''. This map is both linear and invertible, and so defines an element of the [[general linear group]] GL(''E<sub>x</sub>''). The '''holonomy group''' of ∇ based at ''x'' is defined as

:<math>\operatorname{Hol}_x(\nabla) = \{P_\gamma \in \mathrm{GL}(E_x) \mid \gamma \text{ is a loop based at } x\}.</math>

The '''restricted holonomy group''' based at ''x'' is the subgroup <math>\operatorname{Hol}^0_x(\nabla)</math> coming from [[contractible]] loops&nbsp;''γ''.

If ''M'' is [[connected space|connected]], then the holonomy group depends on the [[Pointed space|basepoint]] ''x'' only [[up to]] [[Conjugacy class|conjugation]] in GL(''k'', '''R'''). Explicitly, if ''γ'' is a path from ''x'' to ''y'' in ''M'', then

:<math>\operatorname{Hol}_y(\nabla) = P_\gamma \operatorname{Hol}_x(\nabla) P_\gamma^{-1}.</math>

Choosing different identifications of ''E<sub>x</sub>'' with '''R'''<sup>''k''</sup> also gives conjugate subgroups. Sometimes, particularly in general or informal discussions (such as below), one may drop reference to the basepoint, with the understanding that the definition is good up to conjugation.

Some important properties of the holonomy group include:
* <math>\operatorname{Hol}^0(\nabla)</math> is a connected [[Lie subgroup]] of GL(''k'', '''R''').
* <math>\operatorname{Hol}^0(\nabla)</math> is the [[identity component]] of <math>\operatorname{Hol}(\nabla).</math>
* There is a natural, [[surjective]] [[group homomorphism]] <math>\pi_1(M) \to \operatorname{Hol}(\nabla)/ \operatorname{Hol}^0(\nabla),</math> where <math>\pi_1(M)</math> is the [[fundamental group]] of ''M'', which sends the homotopy class <math>[\gamma]</math> to the [[coset]] <math>P_{\gamma}\cdot\operatorname{Hol}^0(\nabla).</math>
* If ''M'' is [[simply connected]], then <math>\operatorname{Hol}(\nabla) = \operatorname{Hol}^0(\nabla).</math>
* ∇ is flat (i.e. has vanishing curvature) [[if and only if]] <math>\operatorname{Hol}^0(\nabla)</math> is trivial.
<!--Find a better place for this, please:See also [[Wilson loop]].-->